\paragraph{Example.}
Suppose we are given the set of species $\res=\{A, B, C, D, E\}$ with decay function $\life$ such that 
$$
\begin{array}{lcl}
\life(A) &=& 7 \\
\life(B) &=& 2 \\
\life(C) &=& 4 \\
\life(D) &=& 1 \\ 
\life(E) &=& 4 
\end{array}
$$
and a \rsd \RS that contains the following reactions:
$$
\begin{array}{lcll}
r_1 &=&(\{ A, B \},\{C\},\{B,D\}) & \dur(r_1)=3 \\
r_2 &=&(\{E,C\},\{D\},\{A\}) & \dur(r_2)=1 \\
r_3 &=&(\{A\},\{C\},\{B\}) & \dur(r_3)=2 
\end{array}
$$
with initial set $\res_0 =\{B,C,E\}.$




\begin{figure}[t]
\centering
\begin{tikzpicture}[node distance=1.5cm,>=stealth',bend angle=45,auto]

  \tikzstyle{place}=[circle,thick,draw=blue!75,fill=blue!30,minimum size=7mm]
 \tikzstyle{no place}=[circle,thick,draw=blue!0,fill=blue!0,minimum size=7mm]
  \tikzstyle{red place}=[place,draw=red!75,fill=red!20]
  \tikzstyle{transition}=[rectangle,thick,draw=black!75,
  			  fill=black!20,minimum size=4mm]
  \tikzstyle{every token}=[font=\small]



  \node [place] at (4,4) (A){$A$};
  \node [place] at (0,4) (B){$B$};
  \node [place] at (2,4) (C){$C$};
  \node [place] at (6,4) (D){$D$};
  \node [place ] at (8,4) (E){$E$};
  \node [place]  at (10,2) (Pc){$\Pclock$};
  
  \node [transition]  at (1,0) (tr3) {$L(t_{r_3})$}
        edge [pre and post, bend right]                (Pc)
        edge [pre and post]                (A)
        edge [pre and post]                (B)
        edge [pre and post]                (C);

  \node [transition]  at (4,1) (tr1) {$L(t_{r_1})$}
        edge [pre and post, bend right]                (Pc)
        edge [pre and post]                (A)
        edge [pre and post]                (B)
        edge [pre and post]                (C)
        edge [pre and post]                (D);

  \node [transition]  at (7,2) (tr2) {$L(t_{r_2})$}
        edge [pre and post]                (Pc)
        edge [pre and post]                (A)
        edge [pre and post]                (D)
        edge [pre and post]                (C)
        edge [pre and post]                (E);

  \node [transition]  at (4,6) (tc) {$L(t_c)$}
        edge [pre and post, bend left]                (Pc)
        edge [pre and post]                (A)
        edge [pre and post]                (B)
        edge [pre and post]                (C)
        edge [pre and post]                (D)
        edge [pre and post]                (E);


%        edge [pre and post]         node[right] {$\tuple{0,0,\birth_k}$}           (I)
%        edge [pre, bend right]         node[left] {$\tuple{\bool_k,\refr_k,\birth_k}$}        (P)
%        edge [post, bend left]       node[right] {$\tuple{1,z,\birth_k'}$}          (P);
    
\end{tikzpicture}
\caption{Petri net for the reaction system \RS }  \label{fig:example}
\end{figure}


The associated Petri net with timestamps, depicted in Figure \ref{fig:example}, is  defined as the tuple $(Q, T,F, L, m_0)$ where:
\begin{itemize}

 \item $F$ such that:
 $$
\begin{array}{l}
 \pre{t_c} = \post{t_c}= Q \\

\pre{t_{r_1}} =  \post{t_{r_1}}= \{A,B,C,D, \Pclock\} \\
\pre{t_{r_2}} =  \post{t_{r_2}}= \{A,C,D,E, \Pclock\} \\
\pre{t_{r_3}} =  \post{t_{r_3}}= \{A,B,C, \Pclock\} 
\end{array}
 $$


\item Labels for Transitions:
 $$
\begin{array}{lcl}
 L(t_c) &= & \bigwedge_{S_i \in \res} C_i \wedge (z'=z+1) \text{ where }\\
 C_i      &= & \big [ (\bool_i = 1 \wedge  z-\refr_i \leq \life(S_i) \wedge \bool_i' = \bool_i \wedge \refr_i'=\refr_i \wedge \birth_i'=\birth) \, \vee \\
     &&\ (\bool_i = 1 \wedge  z-\refr_i > \life(S_i) \wedge \bool_i' = 0 \wedge \refr_i'=0 \wedge \birth_i'=z') \,  \vee \\
     &&\ (\bool_i = 0  \wedge \bool_i' = \bool_i \wedge \refr_i'=\refr_i \wedge \birth_i'=\birth)\big]\\ \\

 L(t_{r_1}) \! \!&=& (z-\birth_A \geq 3) \wedge (z-\birth_B \geq 3) \wedge (z-\birth_C \geq 3)   \wedge \\
&& (\bool_D=0 \wedge \birth_D' = z) \vee (\bool_D=1 \wedge \birth_D' = \birth_D)  \\ \\

 L(t_{r_2}) \! \!&=& (z-\birth_C \geq 1) \wedge (z-\birth_D \geq 1) \wedge (z-\birth_E \geq 1)   \wedge \\
&& (\bool_A=0 \wedge \birth_A' = z) \vee (\bool_A=1 \wedge \birth_A' = \birth_A)    \\ \\  

 L(t_{r_3}) \! \!&=& (z-\birth_A \geq 2)  \wedge (z-\birth_C \geq 2)   \wedge \\
&& (\bool_B=0 \wedge \birth_B' = z) \vee (\bool_B=1 \wedge \birth_B' = \birth_D)          


\end{array}
 $$


\item Labels for Arcs:
$$
\begin{array}{cllc}
t_c & L(\Pclock, t_c) = z &  L(t_c, \Pclock ) = z'\\
      & L(S_i, t_c) = \tuple{\bool_i, \refr_i, \birth_i} & L(t_c, S_i) = \tuple{\bool_i', \refr_i', \birth_i'} &  \forall S_i\in \res \\ \\

t_{r_1}  & L(\Pclock, t_{r_1}) = z &  L(t_{r_1}, \Pclock ) = z\\
      & L(A,t_{r_1}) = \tuple{1, \refr_A, \birth_A} & L(t_{r_1}, A) = \tuple{1, \refr_A, \birth_A}  \\
      & L(B,t_{r_1}) = \tuple{1, \refr_B, \birth_B} & L(t_{r_1}, B) = \tuple{1, z, \birth_B}  \\
      & L(C,t_{r_1}) = \tuple{0, 0, \birth_C} & L(t_{r_1}, C) = \tuple{0, 0, \birth_C}  \\
      & L(D,t_{r_1}) = \tuple{\bool_D, \refr_D, \birth_D} & L(t_{r_1}, D) = \tuple{1, z, \birth_D'} \\ \\


t_{r_2}  & L(\Pclock, t_{r_2}) = z &  L(t_{r_2}, \Pclock ) = z\\
      & L(A,t_{r_2}) = \tuple{\bool_A, \refr_A, \birth_A} & L(t_{r_2}, A) = \tuple{1, z, \birth_A'}  \\
      & L(C,t_{r_2}) = \tuple{1, \refr_C, \birth_C}          & L(t_{r_2}, C) = \tuple{1, \refr_C, \birth_C}  \\
      & L(D,t_{r_2}) = \tuple{0,0, \birth_D}                    & L(t_{r_2}, D) = \tuple{0,0, \birth_D} \\
      & L(E,t_{r_2}) = \tuple{1, \refr_E, \birth_E} & L(t_{r_2}, B) = \tuple{1, \refr_E, \birth_E}  \\ \\ 


t_{r_3}  & L(\Pclock, t_{r_3}) = z &  L(t_{r_3}, \Pclock ) = z\\
      & L(A,t_{r_3}) = \tuple{1, \refr_A, \birth_A} & L(t_{r_3}, A) = \tuple{1, \refr_A, \birth_A}  \\
      & L(B,t_{r_3}) = \tuple{\bool_B, \refr_B, \birth_B} & L(t_{r_3}, B) = \tuple{1, z, \birth_B'}  \\
      & L(C,t_{r_3}) = \tuple{0, 0, \birth_C} & L(t_{r_3}, C) = \tuple{0, 0, \birth_C}  
\end{array}
$$

 


\end{itemize}
and the other elements are defined as in Definition \ref{def:rs}.

